3.1391 \(\int \frac {(5-x) (2+3 x^2)^{5/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=133 \[ \frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{9800 (2 x+3)^3}-\frac {9 (2643 x+8575) \sqrt {3 x^2+2}}{19600 (2 x+3)}+\frac {789723 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{39200 \sqrt {35}}+\frac {63}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

[Out]

1/9800*(6637+8193*x)*(3*x^2+2)^(3/2)/(3+2*x)^3+1/140*(23+76*x)*(3*x^2+2)^(5/2)/(3+2*x)^5+63/32*arcsinh(1/2*x*6
^(1/2))*3^(1/2)+789723/1372000*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-9/19600*(8575+2643*x)*(
3*x^2+2)^(1/2)/(3+2*x)

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Rubi [A]  time = 0.08, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {811, 813, 844, 215, 725, 206} \[ \frac {(76 x+23) \left (3 x^2+2\right )^{5/2}}{140 (2 x+3)^5}+\frac {(8193 x+6637) \left (3 x^2+2\right )^{3/2}}{9800 (2 x+3)^3}-\frac {9 (2643 x+8575) \sqrt {3 x^2+2}}{19600 (2 x+3)}+\frac {789723 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{39200 \sqrt {35}}+\frac {63}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

(-9*(8575 + 2643*x)*Sqrt[2 + 3*x^2])/(19600*(3 + 2*x)) + ((6637 + 8193*x)*(2 + 3*x^2)^(3/2))/(9800*(3 + 2*x)^3
) + ((23 + 76*x)*(2 + 3*x^2)^(5/2))/(140*(3 + 2*x)^5) + (63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + (789723*ArcTanh
[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(39200*Sqrt[35])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx &=\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}-\frac {\int \frac {(-1248+1752 x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{1120}\\ &=\frac {(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac {\int \frac {(372096-1522368 x) \sqrt {2+3 x^2}}{(3+2 x)^2} \, dx}{627200}\\ &=-\frac {9 (8575+2643 x) \sqrt {2+3 x^2}}{19600 (3+2 x)}+\frac {(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}-\frac {\int \frac {12178944-59270400 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{5017600}\\ &=-\frac {9 (8575+2643 x) \sqrt {2+3 x^2}}{19600 (3+2 x)}+\frac {(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac {189}{32} \int \frac {1}{\sqrt {2+3 x^2}} \, dx-\frac {789723 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{39200}\\ &=-\frac {9 (8575+2643 x) \sqrt {2+3 x^2}}{19600 (3+2 x)}+\frac {(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac {63}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {789723 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{39200}\\ &=-\frac {9 (8575+2643 x) \sqrt {2+3 x^2}}{19600 (3+2 x)}+\frac {(6637+8193 x) \left (2+3 x^2\right )^{3/2}}{9800 (3+2 x)^3}+\frac {(23+76 x) \left (2+3 x^2\right )^{5/2}}{140 (3+2 x)^5}+\frac {63}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+\frac {789723 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{39200 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 100, normalized size = 0.75 \[ \frac {789723 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {70 \sqrt {3 x^2+2} \left (88200 x^5+2740188 x^4+11367738 x^3+20911298 x^2+17940463 x+5999363\right )}{(2 x+3)^5}}{1372000}+\frac {63}{32} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6,x]

[Out]

(63*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/32 + ((-70*Sqrt[2 + 3*x^2]*(5999363 + 17940463*x + 20911298*x^2 + 11367738*x
^3 + 2740188*x^4 + 88200*x^5))/(3 + 2*x)^5 + 789723*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/13
72000

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fricas [A]  time = 0.78, size = 191, normalized size = 1.44 \[ \frac {2701125 \, \sqrt {3} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 789723 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 140 \, {\left (88200 \, x^{5} + 2740188 \, x^{4} + 11367738 \, x^{3} + 20911298 \, x^{2} + 17940463 \, x + 5999363\right )} \sqrt {3 \, x^{2} + 2}}{2744000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/2744000*(2701125*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log(-sqrt(3)*sqrt(3*x^2 + 2)*
x - 3*x^2 - 1) + 789723*sqrt(35)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log((sqrt(35)*sqrt(3*x^
2 + 2)*(9*x - 4) - 93*x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 140*(88200*x^5 + 2740188*x^4 + 11367738*x^3 + 209
11298*x^2 + 17940463*x + 5999363)*sqrt(3*x^2 + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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giac [B]  time = 0.32, size = 355, normalized size = 2.67 \[ -\frac {63}{32} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {789723}{1372000} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9}{64} \, \sqrt {3 \, x^{2} + 2} - \frac {3 \, \sqrt {3} {\left (1034487 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 28143036 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 94364251 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 328235733 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 120044232 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 774358774 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 578739476 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 495467552 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 66595728 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 11086336\right )}}{156800 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

-63/32*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 789723/1372000*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) -
3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) - 9/64*sqrt(3*x^2 + 2
) - 3/156800*sqrt(3)*(1034487*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^9 + 28143036*(sqrt(3)*x - sqrt(3*x^2 + 2))
^8 + 94364251*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 328235733*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 120044232*
sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 - 774358774*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 + 578739476*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 2))^3 - 495467552*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 66595728*sqrt(3)*(sqrt(3)*x - sqrt(3*x
^2 + 2)) - 11086336)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^5

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maple [B]  time = 0.06, size = 248, normalized size = 1.86 \[ \frac {1131399 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}} x}{525218750}+\frac {267723 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{12005000}+\frac {248967 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{686000}+\frac {63 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{32}+\frac {789723 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{1372000}-\frac {11 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{24500 \left (x +\frac {3}{2}\right )^{4}}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}-\frac {521 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{857500 \left (x +\frac {3}{2}\right )^{3}}-\frac {2241 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{30012500 \left (x +\frac {3}{2}\right )^{2}}-\frac {377133 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{525218750 \left (x +\frac {3}{2}\right )}-\frac {789723 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{262609375}-\frac {263241 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6002500}-\frac {789723 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1372000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(5/2)/(2*x+3)^6,x)

[Out]

-11/24500/(x+3/2)^4*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-13/5600/(x+3/2)^5*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-521/857500/(
x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(7/2)-2241/30012500/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+1131399/525218750
*(-9*x+3*(x+3/2)^2-19/4)^(5/2)*x-377133/525218750/(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(7/2)+267723/12005000*(-9*x+
3*(x+3/2)^2-19/4)^(3/2)*x+248967/686000*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x+789723/1372000*35^(1/2)*arctanh(2/35*(
-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+63/32*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-789723/262609375*(-9*x+3*
(x+3/2)^2-19/4)^(5/2)-263241/6002500*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-789723/1372000*(-36*x+12*(x+3/2)^2-19)^(1/2
)

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maxima [B]  time = 1.24, size = 244, normalized size = 1.83 \[ \frac {6723}{30012500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {44 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{6125 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1042 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{214375 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2241 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{7503125 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {267723}{12005000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x - \frac {263241}{6002500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {377133 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{30012500 \, {\left (2 \, x + 3\right )}} + \frac {248967}{686000} \, \sqrt {3 \, x^{2} + 2} x + \frac {63}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {789723}{1372000} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {789723}{686000} \, \sqrt {3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

6723/30012500*(3*x^2 + 2)^(5/2) - 13/175*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 24
3) - 44/6125*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1042/214375*(3*x^2 + 2)^(7/2)/(8*x^3
+ 36*x^2 + 54*x + 27) - 2241/7503125*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 267723/12005000*(3*x^2 + 2)^(3/2)
*x - 263241/6002500*(3*x^2 + 2)^(3/2) - 377133/30012500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 248967/686000*sqrt(3*x^2
+ 2)*x + 63/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 789723/1372000*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) -
2/3*sqrt(6)/abs(2*x + 3)) - 789723/686000*sqrt(3*x^2 + 2)

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mupad [B]  time = 1.95, size = 206, normalized size = 1.55 \[ \frac {63\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {9\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{64}-\frac {789723\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1372000}+\frac {789723\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1372000}+\frac {2303\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{512\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {3185\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2048\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {64959\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{19600\,\left (x+\frac {3}{2}\right )}+\frac {44127\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8960\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {15397\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{2560\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^2 + 2)^(5/2)*(x - 5))/(2*x + 3)^6,x)

[Out]

(63*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/32 - (9*3^(1/2)*(x^2 + 2/3)^(1/2))/64 - (789723*35^(1/2)*log(x + 3/2
))/1372000 + (789723*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1372000 + (2303*3^(1/2)*(
x^2 + 2/3)^(1/2))/(512*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (3185*3^(1/2)*(x^2 + 2/3)^(1/2))/(2048
*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32)) - (64959*3^(1/2)*(x^2 + 2/3)^(1/2))/(196
00*(x + 3/2)) + (44127*3^(1/2)*(x^2 + 2/3)^(1/2))/(8960*(3*x + x^2 + 9/4)) - (15397*3^(1/2)*(x^2 + 2/3)^(1/2))
/(2560*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(5/2)/(3+2*x)**6,x)

[Out]

Timed out

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